^{1, 2}

^{2}

^{1}

^{2}

We investigate the local stability, prime period-two
solutions, boundedness, invariant intervals, and global attractivity
of all positive solutions of the following difference equation:

Our aim in this paper is to study the dynamical behavior of the following rational difference equation

When

In [

Let

The main purpose of this paper is to further consider the global attractivity of all positive solutions of (

For the general theory of difference equations, one can refer to the monographes [

For the sake of convenience, we firstly present some definitions and known results which will be useful in the sequel.

Let

A point

That is,

An interval

That is, every solution of (

Let

denote the partial derivatives of

Assume that

The following result will be useful in establishing the global attractivity character of the equilibrium of (

Suppose that a continuous function

(Note that for

Then (

This work is organized as follows. In Section

The unique positive equilibrium of (

The linearized equation associated with (

and its characteristic equation is

From this and Lemma

Assume that

If

If

If

In the following, we will consider the period-two solutions of (

Let

be a period-two solution of (

If

then

If

then

From the above discussion, we have the following result.

Equation (

In this section, we discuss the boundedness, invariant intervals of (

All positive solutions of (

Equation (

Let

When

When

Set

Assume that

assume

assume

By calculating the partial derivatives of the function

In this subsection, we discuss the invariant interval of (

Assume that

If for some

If for some

If for some

If

If

If

The proofs of (i)–(iv) are straightforward consequences of the identities (

(v) Using the decreasing character of

On the other hand,

(vi) By using the monotonic character of

On the other hand, similar to (v) it can be proved that

(vii) In this case note that

When

Assume that

If for some

If for some

If for some

Assume that

If for some

If for some

If for some

If

If

If

The proofs of (i)–(iv) are direct consequences of the identities (

(v) Using the decreasing character of

On the other hand, similar to Lemma

(vi) By using the monotonic character of

On the other hand, similar to Lemma

(vii) In this case note that

Furthermore, similar to Lemma

Assume that

If for some

If for some

If for some

If for some

If for some

In this case, we have that

In this section, we discuss the global attractivity of the positive equilibrium of (

In this subsection, we discuss the behavior of positive solutions of (

Assume that

By the change of variables

In this subsection, we present global attractivity of (

The following result is straightforward consequence of the identity (

Assume that

Suppose that

Suppose that

Assume that

Suppose

Suppose

We only give the proof of (i), the proof of (ii) is similar and will be omitted. First, note that in this case

If

Now, to complete the proof it remains to show that when

If

Case (a). From Lemma

If

So assume for the sake of contradiction, that for all

Set

Assume that

To complete the proof, there are four cases to be considered.

Case (i).

By Theorem

Case (ii).

In this case, the only positive equilibrium is

Let

Set

Case (iii).

By Theorem

Case (iv).

In this case, we note that

In this subsection, we discuss the global behavior of (

The following three results are the direct consequences of equations (

Assume that

If for some

If for some

If for some

Assume that

If for some

If for some

If for some

Assume that

If for some

If for some

Assume that

If

If

If

We only give the proof of (i), the proofs of (ii) and (iii) are similar and will be omitted.

When

Case (i). If for some

Case (ii). If for some

Case (iii). If for some

Assume that

The proof will be accomplished by considering the following four cases.

Case (i).

By part (i) of Theorem

Case (ii).

In this case, the only positive equilibrium of (

Case (iii).

By Theorem

Case (iv).

In this case, we note that

Finally, we summarize our results and obtain the following theorem, which shows that

The unique positive equilibrium